Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.0.788 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}17&0\\52&19\end{bmatrix}$, $\begin{bmatrix}35&52\\44&1\end{bmatrix}$, $\begin{bmatrix}53&17\\12&53\end{bmatrix}$, $\begin{bmatrix}55&53\\20&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.k.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $64512$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 49 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2}\cdot\frac{x^{24}(x^{4}-16x^{3}y-112x^{2}y^{2}-128xy^{3}+64y^{4})^{3}(x^{4}+16x^{3}y-112x^{2}y^{2}+128xy^{3}+64y^{4})^{3}}{y^{2}x^{26}(x^{2}-8y^{2})^{2}(x^{2}+8y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.24.0-8.n.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ |
| 56.24.0-8.n.1.8 | $56$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 56.96.0-8.m.1.2 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.m.1.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.m.2.1 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-8.m.2.4 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bc.1.4 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bc.1.5 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bc.2.3 | $56$ | $2$ | $2$ | $0$ |
| 56.96.0-56.bc.2.6 | $56$ | $2$ | $2$ | $0$ |
| 56.384.11-56.bt.1.10 | $56$ | $8$ | $8$ | $11$ |
| 56.1008.34-56.cp.1.22 | $56$ | $21$ | $21$ | $34$ |
| 56.1344.45-56.cr.1.19 | $56$ | $28$ | $28$ | $45$ |
| 112.96.0-16.e.1.4 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-16.e.1.7 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.e.1.10 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.e.1.15 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-16.f.1.6 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-16.f.1.7 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.f.1.11 | $112$ | $2$ | $2$ | $0$ |
| 112.96.0-112.f.1.14 | $112$ | $2$ | $2$ | $0$ |
| 112.96.1-16.d.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-16.d.1.7 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.d.1.3 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.d.1.13 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-16.f.1.2 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-16.f.1.7 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.f.1.5 | $112$ | $2$ | $2$ | $1$ |
| 112.96.1-112.f.1.11 | $112$ | $2$ | $2$ | $1$ |
| 168.96.0-24.bd.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bd.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bd.2.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bd.2.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cz.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cz.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cz.2.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.cz.2.14 | $168$ | $2$ | $2$ | $0$ |
| 168.144.4-24.cp.1.4 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-24.cr.1.8 | $168$ | $4$ | $4$ | $3$ |
| 280.96.0-40.be.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.be.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.be.2.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-40.be.2.7 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.da.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.da.1.15 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.da.2.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.da.2.15 | $280$ | $2$ | $2$ | $0$ |
| 280.240.8-40.z.1.9 | $280$ | $5$ | $5$ | $8$ |
| 280.288.7-40.bx.1.24 | $280$ | $6$ | $6$ | $7$ |
| 280.480.15-40.cp.1.10 | $280$ | $10$ | $10$ | $15$ |